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Rules for differentiation (4)

Product Rule

Let \(f(x)\) and \(g(x)\) be differentiable functions. Then

\[ \large \frac{d}{dx}[f(x)*g(x)]=f(x)g'(x)+g(x)f'(x) \]

What this is saying:

• Breakdown multiplication into two separate functions

• Then take the first function times the derivative of the second, and add the second function times the derivative of the first.

Product rule example

✏️ Find the derivative of \(y=2x(x^2+3x)\).

Product rule example

✏️ Find the derivative of \(y=2x(x^2+3x)\).

We have that:

\[ \begin{align} y&=2x(x^2+3x) \\[3px] y&=\overbrace{2x}^{f(x)}\overbrace{(x^2+3x)}^{g(x)} &\text{Break into two functions}\\[3px] \frac{dy}{dx}&=\overbrace{2x}^{f(x)}\overbrace{(2x+3)}^{g'(x)}+\overbrace{2}^{f'(x)}\overbrace{(x^2+3x)}^{g(x)} &\text{Apply Power Rule}\\[3px] \frac{dy}{dx}&=6x^2+12x &\text{Algebra clean up} \end{align} \]

What is another way we could have computed this derivative?

Rules for differentiation (5)

Quotient Rule

\[ \large \frac{d}{dx}\left [\frac{f(x)}{g(x)}\right]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2} \]

Notice:

• Same idea as product rule, but order matters.

• The denominator has to be \(g(x)\)

Quotient rule example

✏️ Find the derivative of \(y=\frac{x^2}{(x+1)}\).

Quotient rule example

✏️ Find the derivative of \(y=\frac{x^2}{(x+1)}\).

\[ \begin{align} y &= \frac{x^2}{(x+1)} \\[3px] y &= \frac{\overbrace{x^2}^{f(x)}}{\underbrace{(x+1)}_{g(x)}} &\text{Identify functions}\\[3px] \frac{dy}{dx} &= \frac{\overbrace{(x+1)}^{g(x)}\overbrace{2x}^{f'(x)}-\overbrace{x^2}^{f(x)}\overbrace{(1)}^{g'(x)}}{\underbrace{(x+1)^2}_{[g(x)]^2}} &\text{Quotient rule}\\[3px] \frac{dy}{dx} &= \frac{x^2+2x}{(x+1)^2} &\text{Simplify} \\[3px] \end{align} \]

Note: Applying the quotient rule can yield some complex algebraic expressions. Try to reduce it down to a usable form.

Composition of functions

If we have two functions \(f\) and \(g\), the composition of \(f\) and \(g\) is

\[f\circ g (x) = f(g(x)).\]

This means that the output of \(g(x)\) is now the input of \(f(x)\).

Composition of functions means applying one function to the result of another.

Composition of functions

If we have two functions \(f\) and \(g\), the composition of \(f\) and \(g\) is

\[f\circ g (x) = f(g(x)).\]

This means that the output of \(g(x)\) is now the input of \(f(x)\).

Composition of functions means applying one function to the result of another.

Example

✏️ Let \(f(x) = x^2\) and \(g(x) = 2x+3\). What is \(f\circ g (x)\)?

We have that

\[f\circ g (x) = (2x+3)^2.\]

Example

✏️ Let \(h(x) = \sqrt{ 3x^2 + 10}\). How can we write \(h\) as a composition of functions?

If we make \(f(x) = \sqrt{x}\) and \(g(x) = 3x^2 +10\) then we have that

\[ f\circ g (x) = \sqrt{ 3x^2 + 10}.\]

Rules for differentiation (6)

Chain Rule

\[ \large \frac{d}{dx}f\circ g(x)= \frac{d}{dx} f(g(x)) = f'(g(x))\cdot g'(x) \]

What this is saying:

• Take the derivative of the outermost function then multiply by the derivative of the innermost.

• Probably the most important rule for advanced calculus!

Chain rule example

✏️ What is the derivative of \(y = (x^3 -2)^2\)? Let’s use a trick called \(u\)-substitution.

Chain rule example

✏️ What is the derivative of \(y = (x^3 -2)^2\)? Let’s use a trick called \(u\)-substitution.

1. Identify the inner function and label it as \(u\). Rewrite \(y\) in terms of \(u\):

\[ \begin{align} y &= (x^3-2)^2 \\ u &= x^3-2 \\ y &= u^2 \end{align} \]

2. Take the derivative of \(y\) with respect to \(u\) and substitute \(u\):

\[\frac{dy}{du} = 2u = 2(x^3-2).\]

3. And the derivative of \(u\) with respect to \(x\):

\[\frac{du}{dx} = 3x^2.\]

4. Put everything back into the chain rule and simplify:

\[ \begin{align} \frac{dy}{dx} &= \frac{dy}{du}\cdot\frac{du}{dx} \\ &= 2(x^3-2)(3x^2) = 6x^5 -12x^2. \end{align} \]

The number \(e\) and exponential funcitons

The number \(e\) is a constant in mathematics, it is approximately \(e\approx 2.72\), but really its decimals continue infinitely without any pattern.

Derivative of the exponential function

We have that \[ \large \frac{d}{dx}e^x=e^x \]

The derivative at every point on this line is simply \(e(x)\).

Derivative of the exponential function

Higher order derivatives

• We can continue to take derivatives of derivatives

• The second derivative becomes the rate of change of the rate of change (Think acceleration)

• Notation:

  • \(f''(x)\) or \(\frac{d^2y}{dx^2}\): 2nd Derivative

  • \(f^{(n)}(x)\) or \(\frac{d^ny}{dx^n}\) nth derivative

Higher order derivatives

✏️ Identify which rule or rules works best for each function.

\[ \begin{align} \text{A) }y=e^{\sqrt{3x+10}} & &\text{B) }g(y)=\frac{(8y+2)}{(4y+3)^3} & & \text{C) }p(b)=18b\cdot e^{2b} \end{align} \]

1. Find the first derivative of

\[ f(x)=(2x-x^2)^3. \]

2. Find the first derivative of

\[ f(x)=\frac{2}{x^2+1}. \]

3. Find the third derivative of

\[ f(x)=x^2(2x+1). \]

Exponential functions in the world

Infectious diseases

Human population

Communication

Growth of Renewable Energy

Logistic growth

Growth modeled by an exponential function is unbounded, but resource limits do exist.

Modeling growth using a logistic growth model: growth is initially exponential but levels off as the environment’s carrying capacity is reached. The growth rate is given by

\[ \frac{dN}{dt} = rN\left(\frac{K-N}{K}\right). \]

where \(N=\text{population, }r=\text{growth rate, }t=\text{time, and }K=\text{carrying capacity}.\)

Extremely important in fisheries and biological Ccnservation

Using derivatives we can find crucial information to solve environmental problems

Derivatives allow us to:

1. Find instantaneous rate of change at any point

2. Find when functions are increasing or decreasing

3. Find the optimal values of functions

   • Optimal values are key to designing effective policies

How quickly is the human population rising right now?

Dochy et al. (1995) used the following function to describe the increase in human population over time:

\[ x(t)=\frac{1}{c-\beta t} \]

Where \(c=9.833*10^{-3}\), \(\beta=4.849*10^{-6}\), and \(x\) is the total human population in millions at year \(t\).

Use derivatives to find the answers!

- How quickly was the human population increasing in 1450?

The model isn’t wrong, just not predictive out of sample

Increasing and decreasing functions

A function is increasing on an interval \((a,b)\) is \(f(x_1) < f(x_2)\) every time \(x_1<x_2\).

A function is decreasing on an interval \((a,b)\) is \(f(x_1) > f(x_2)\) every time \(x_1<x_2\).

A function is constant on an interval \((a,b)\) is \(f(x_1) = f(x_2)\) for every \(x_1, x_2\) in \((a,b)\).

🙂 This all really means what you think it means:

Increasing and decreasing functions

Let \(f(x)\) be a function that is differentiable on the interval \((a,b)\), then:

• If \(f'(x)>0\) for all \(x\) values between \(a\) and \(b\), \(f\) is increasing on \((a,b)\)

• If \(f'(x)<0\) for all \(x\) values betwen \(a\) and \(b\), this interval is said to be decreasing

• If \(f'(x)=0\) for all values between \(a\) and \(b\), then the function is constant within the interval

Optimization problems find relative extremum (aka max or min) of functions

1. Find First Order Conditions:

• Relative extremum exist at all points on function where \(f'(x)=0\)

Graphical Example of Optimization

Solve Analytically

First Order Conditions:

\[ \begin{align} f(x)&=x^3-12x+2 \\ f'(x)&=3x^2-12 \\ x^*&=-2 & x^*=2 \end{align} \]

1. How many local min and maximums can you find on this graph?

2. Solve for all optimum. Identify which are minima and which are maxima.

\[ f(x)=2x^3-6x^2+3x+15 \]

3. Your team has been hired to help the city of Santa Barbara to help eliminate a nasty pollutant in our streams. You are choosing between two policies. Policy A provides \(\sqrt{10x+4}\) benefits where \(x\) is the dollar spent on interventions. Policy B provides \(\sqrt{20x+4}\) benefits for every dollar spent. However, each policy has a cost associated with each. Policy A has a constant linear cost at \(2x\) and Policy B has a constant linear cost at \(3x\).

1. Find the optimal amount of interventions dollars to spend for each policy. Consider both costs and benefits. [hint: Remember \(\sqrt{}\) is just \(x^{\frac{1}{2}}\). You can use exponent rules to help solve]

2. Which policy provides the most benefits?

3. Which policy is cheaper?

4. Which policy should the city of Santa Barbara pursue?